Abstract. Let X1, X2, . . . denote i.i.d. random bits, each taking the values 1 and 0 with respective probabilities p and 1 − p. A well-known theorem of Erdős and Rényi (1970) describes the length of the longest contiguous stretch, or “run,” of ones in X1, . . . , Xn for large values of n. Benjamini, Häggström, Peres, and Steif (2003, Theorem 1.4) demonstrated the existence of unusual times, pr...

N lim 1( 1: f(nkx)) = 0, N-N k_l or roughly speaking the strong law of large numbers holds for f(nkx) (in fact the authors prove that Ef(nkx)/k converges almost everywhere) . The question was raised whether (2) holds for any f(x) . This was known for the case nk=2k( 2) . In the present paper it is shown that this is not the case . In fact we prove the following theorem . THEOREM 1 . There exist...

This paper extends the author’s earlier work on the Law of Large Numbers for fuzzy numbers [2] to the case where the fuzzy numbers are of type L-R. Namely, we shall define a class of Archimedean triangular norms in which the equality lim n→∞ Nes(mn − ≤ ηn ≤ mn + = 1, for any > 0, holds for all sequences of fuzzy numbers, ξi = (Mi, α, β)LR, i ∈ N, with twice differentiable and concave shape func...

We study the following problem: If ξ1, ξ2, . . . are fuzzy numbers with modal values M1, M2, . . . , then what is the strongest t-norm for which lim n→∞ Nes ( mn − ≤ ξ1 + · · ·+ ξn n ≤ mn + ) = 1, for any > 0, where mn = M1 + · · ·+Mn n , the arithmetic mean ξ1 + · · ·+ ξn n is defined via sup-t-norm convolution and Nes denotes necessity.

in this paper, we generalize some results of chandra and goswami [4] for pairwise negatively dependent random variables (henceforth r.v.’s). furthermore, we give baum and katz’s [1] type results on estimate for the rate of convergence in these laws.

We discuss in this paper the strong convergence for weighted sums of negatively orthant dependent (NOD) random variables by generalized Gaussian techniques. As a corollary, a Cesaro law of large numbers of i.i.d. random variables is extended in NOD setting by generalized Gaussian techniques.